An official website of the United States government
An important sparse tensor computation is sparse-tensor-dense-matrix multiplication (SpTM), which is used in tensor decomposition and applications. SpTM is a multi-dimensional analog to sparse-matrix-dense-matrix multiplication (SpMM). In this article, we employ a hierarchical tensor data layout that can unfold a multidimensional tensor to derive a 2D matrix, making it possible to compute SpTM using SpMM kernel implementations for GPUs. We compare two SpMM implementations to the state-of-the-art PASTA sparse tensor contraction implementation using: (1) SpMM with hierarchical tensor data layout; and, (2) unfolding followed by an invocation of cuSPARSE’s SpMM. Results show that SpMM can outperform PASTA 70.9% of the time, but none of the three approaches is best overall. Therefore, we use a decision tree classifier to identify the best performing sparse tensor contraction kernel based on precomputed properties of the sparse tensor.
more »
« less
Fourier–Mukai transforms commuting with Frobenius
We show that a Fourier–Mukai equivalence between smooth projective varieties of characteristic that commutes with either pushforward or pullback along Frobenius is a composition of shifts, isomorphisms, and tensor products with invertible sheaves whose th tensor power is trivial.
more »
« less
- Award ID(s):
- 1840190
- PAR ID:
- 10616230
- Publisher / Repository:
- Wiley
- Date Published:
- Journal Name:
- Bulletin of the London Mathematical Society
- Volume:
- 56
- Issue:
- 11
- ISSN:
- 0024-6093
- Page Range / eLocation ID:
- 3477 to 3483
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
null (Ed.)Real-world spatio-temporal data is often incomplete or inaccurate due to various data loading delays. For example, a location-disease-time tensor of case counts can have multiple delayed updates of recent temporal slices for some locations or diseases. Recovering such missing or noisy (under-reported) elements of the input tensor can be viewed as a generalized tensor completion problem. Existing tensor completion methods usually assume that i) missing elements are randomly distributed and ii) noise for each tensor element is i.i.d. zero-mean. Both assumptions can be violated for spatio-temporal tensor data. We often observe multiple versions of the input tensor with different under-reporting noise levels. The amount of noise can be time- or location-dependent as more updates are progressively introduced to the tensor. We model such dynamic data as a multi-version tensor with an extra tensor mode capturing the data updates. We propose a low-rank tensor model to predict the updates over time. We demonstrate that our method can accurately predict the ground-truth values of many real-world tensors. We obtain up to 27.2% lower root mean-squared-error compared to the best baseline method. Finally, we extend our method to track the tensor data over time, leading to significant computational savings.more » « less
-
With the advent of machine learning and its overarching pervasiveness it is imperative to devise ways to represent large datasets efficiently while distilling intrinsic features necessary for subsequent analysis. The primary workhorse used in data dimensionality reduction and feature extraction has been the matrix singular value decomposition (SVD), which presupposes that data have been arranged in matrix format. A primary goal in this study is to show that high-dimensional datasets are more compressible when treated as tensors (i.e., multiway arrays) and compressed via tensor-SVDs under the tensor-tensor product constructs and its generalizations. We begin by proving Eckart–Young optimality results for families of tensor-SVDs under two different truncation strategies. Since such optimality properties can be proven in both matrix and tensor-based algebras, a fundamental question arises: Does the tensor construct subsume the matrix construct in terms of representation efficiency? The answer is positive, as proven by showing that a tensor-tensor representation of an equal dimensional spanning space can be superior to its matrix counterpart. We then use these optimality results to investigate how the compressed representation provided by the truncated tensor SVD is related both theoretically and empirically to its two closest tensor-based analogs, the truncated high-order SVD and the truncated tensor-train SVD.more » « less
-
This work introduces a mathematical approach to analysing the polymer dynamics in turbulent viscoelastic flows that uses a new geometric decomposition of the conformation tensor, along with associated scalar measures of the polymer fluctuations. The approach circumvents an inherent difficulty in traditional Reynolds decompositions of the conformation tensor: the fluctuating tensor fields are not positive definite and so do not retain the physical meaning of the tensor. The geometric decomposition of the conformation tensor yields both mean and fluctuating tensor fields that are positive definite. The fluctuating tensor in the present decomposition has a clear physical interpretation as a polymer deformation relative to the mean configuration. Scalar measures of this fluctuating conformation tensor are developed based on the non-Euclidean geometry of the set of positive definite tensors. Drag-reduced viscoelastic turbulent channel flow is then used an example case study. The conformation tensor field, obtained using direct numerical simulations, is analysed using the proposed framework.more » « less
-
A major challenge in many machine learning tasks is that the model expressive power depends on model size. Low-rank tensor methods are an efficient tool for handling the curse of dimensionality in many large-scale machine learning models. The major challenges in training a tensor learning model include how to process the high-volume data, how to determine the tensor rank automatically, and how to estimate the uncertainty of the results. While existing tensor learning focuses on a specific task, this paper proposes a generic Bayesian framework that can be employed to solve a broad class of tensor learning problems such as tensor completion, tensor regression, and tensorized neural networks. We develop a low-rank tensor prior for automatic rank determination in nonlinear problems. Our method is implemented with both stochastic gradient Hamiltonian Monte Carlo (SGHMC) and Stein Variational Gradient Descent (SVGD). We compare the automatic rank determination and uncertainty quantification of these two solvers. We demonstrate that our proposed method can determine the tensor rank automatically and can quantify the uncertainty of the obtained results. We validate our framework on tensor completion tasks and tensorized neural network training tasks.more » « less
